fracture mechanics experimental tests (astm standards...

Task 6 - Safety Review and LicensingOn the Job Training on Stress Analysis

Pisa (Italy)June 15 – July 14, 2015

Fracture Mechanics experimental tests (ASTM standards and data) 2/2

Davide Mazzini – Ciro Santus

2

Content

• Fracture Toughness KIc

- Plane strain condition

- ASTM standard E399

• High Toughness JIc

- Limitation of the KIc

- ASTM standard E1820

• Measurement of Fatigue Crack Growth Rates

- Paris curve experimental determination, ASTM standard E647

Standard for Fracture Mechanics

Pisa, June 15 – July 14, 2015

3

Wide plasticity CT specimen

Large Fracture Toughness

A very large specimenwould be required to test according to KIc


4

Definition

Rice 1968, J - integral

Let us assume an alternative material with Elastic behavior, not linear (hyperelastic material) and the same Stress/ Strain curve than the actual Elastic-Plastic one

For a monotonic loading the Stress/ Strain history is the same


5

Definition


J is an integral along a path.It does not depend on the path that can be arbitrary provided that it is around the tip of the crack from side to side.

x is the crack direction


6

Definition


J does not depend on the path that can be arbitrary provided that it is around the tip of the crack from side to side.

Zero integral for a closed path

Zero integral for free surfaces

*1 2 3 4

*1 2

1 2

+ + + 0

+ 0=


7

Energy Release Rate

J - integral equivalences

For an Elastic behavior, even not linear, the J integral equals the Energy Release Rate

Potential energy:

U is the stored strain energyF is the work done by external forces


Energy release rate:same as the G (elastic) parameter,A is the crack area.

8

The Griffith problem


Infinite plate with a crack, with linear elastic material, and plane stress,Griffith found:


9

The Griffith problem


Under these conditions:2d

daJ

A E

Being for this problem:

I , 1.0K F a F

Then it follows (that’s the reason of π):2IKJ

E


10

General case


2

where:planestress

planestrain1

Shear modulus2(1 )

E EEE

EG

Only if the material is linear:

2 2 2I II III

2K K KJ

E G


11

CTOD – Crack Tip Opening Displacement

CMOD – Crack Mouth Opening Displacement

COD - Crack Opening Displacement

CTOD

CMOD is just the clip gauge measurement


12

CTOD – Crack Tip Opening Displacement


From ASTM standard E1820


13

CTOD – definition


Alternative way to define the CTOD:2×45° lines

Crack tip blunting(approx. circumferential)


14

CTOD can be related to the SIF under Small Scale Yielding

CTOD under SSY hypothesis


15

CTOD – can also be related to the J integral well beyond the validity

limits of LEFM

CTOD under large plasticity

Path around the strip-yield zone ahead of a crack tip

Plane stress conditions and a nonhardeningmaterial. More generally:

m is a dimensionless constant that depends on the stress state and material properties


16

CTOD – similar value under SSY hypothesis

CTOD under large plasticity

2I

YS

2I

YS

YS

Planestress:4

being:4

approximately:4 1 ( 1)

KE

K JJE

J m


17

J and CTOD

ASTM standard E1820

J and CTOD are still representative even over the 80% of the fully plastic load (plastic collapse)


18

ASTM standard E1820


19

From J back to K

ASTM standard E1820

2

where:planestress

planestress1

Shear modulus2(1 )

E EEE

EG

2 2 2I II III

2K K KJ

E G

ASTM plane strain assumption, mode I only


20

Similar to the previous standard…

ASTM standard E1820

The displacement gage


21


ASTM standard E1820

Types of preparation notches


22


ASTM standard E1820

The CT specimen

The DCT specimen

The SEB specimen


23

The CT specimens

ASTM standard E1820

Two CT specimen types,the increased size for the pin seat is for very high tension loads


24Pisa, June 15 – July 14, 2015

Displacement gage fixture details

ASTM standard E1820

Previous standard

25

J calculation for CT specimen

ASTM standard E1820

Elastic term

Plastic term


26

J calculation for CT specimen

ASTM standard E1820

These values are for the initial crack, just after the (fatigue) precrack


27

J-R curve determination

ASTM standard E1820

a

The suggested procedure is to determine a series of J values for increasingly crack size, and then obtain a (fit) Resistance curve

a


28


ASTM standard E1820

For each crack increment (i) there is an unload cycle to determine the crack size through the compliance

( ) ( ),i iJ a

0 0.5 1 1.5 2 2.5 30

10

20

30

40

50

60

70

Total displacement, mmLo

ad, k

N


29


ASTM standard E1820

(i) stands for the updated values for the crack increments

(i=0) is just the initial crack

K(i) is calculated exactly as the previous standard


30


ASTM standard E1820


31


ASTM standard E1820

Compliance Load Line calculation:


32


ASTM standard E1820

Crack size calculation (compliance method):


33


ASTM standard E1820

Compliance correction


34

J-R curve

ASTM standard E1820

Power law regression line

( ) ( )

, ,

are the regression coefficientsfor fitting the points

,

oq

i i

a B C

J a

Least squares fit procedure to find the regression coefficients


35

J-R curve

ASTM standard E1820


36

J-R curve

ASTM standard E1820

Ic is defined near the initiation of stable crack growth. The precise point is usually ill-defined an offset (similar to yield strength) is required

J


37

J-R curve

ASTM standard E1820

2

Y

Y

2

Ic Ic

Example:kJ N N200 200000 200 200MPa mmm m mm

800MPa

10 2.5mm ...easy to be satisfied!

Conversion:/ (1 ) 220GPa

6630MPa mm 210MPa m

Q

Q

J

JB

E E

K E J

Yield and Ultimateaverage


38

Alternatively the δ-R curve (CTOD-R curve)

ASTM standard E1820

CTOD instead of J


39

J-R curve

ASTM standard E1820

IcIc Ic

Ic is just theonset of fracture,thespecimen can sustain highervaluesof , indeed thecurvecontinues with a (stable)increaseof thecrack size.

QJJ J KE

J

J


40

J-R curve

The Resistance curve

IcRising curveafter J


41

J-R curve


IcRising curveafter J

Stable further propagation


42

J-R curve


1 2

3

1 5

4 unst

paths: Load control, stablecrack growthlimit stable crack

crack

paths: Displacementcontrol paths

stable crack g

able

rowth

i

i

PP

P

PP

4P

Load control is usually less stable than displacement control,in most structures the conditions are between the extremes of load and displacement control


43

J-R curve


Three stages of crack growthin an infinite body

Steady State


44

R curve, single fracture toughness value


Unstable,K J

Ic Ic,K J

crack size a0a

( ), ( )K a J aStable

With a single value the curve is just asymptotic


45

ASTM standard E1820

Homework:

Apply the ASTM E1820 procedure extracting data from the test file:Test J_Ic.xlsx

0 0.5 1 1.5 2 2.5 30

10

20

30

40

50

60

70

Total displacement, mm

Load

, kN

Then estimate JQ and verify if it can be converted into JIc

?


46

The damage tolerant approach

Crack (stable) propagation under fatigue

The presence of a crack (actually detected or just postulated) can be tolerated if the propagation rate is reliably estimated.

The size of the postulated crack is the minimum detectable of the inspection method.


47

Paris’ law (also known as the Paris-Erdogan law)

Fatigue crack propagation

max min

min

max

dd

where:

, are parameters depending on:- the material

-the load ratio

ma C KN

K K K

C m

KRK

Paul C. Paris

I(usually )K K

Time

1N 2N ...N


48

Paris’ law validity range


thKcK

Paris validity

Near threshold

Sudden(unstable) fracture

th is the thresholdstress intensity factor rangebelow this amplitude thecrack remains same sizethough fatigue loaded

K

max

c

Ic

is the thresholdstress intensity factor rangefor which Approaching this valuethecrack turns intounstable propagation

K K

K


49

Paris’ law integration


1-2 /2 ( 2)/2 ( 2)/21 2

d ( )d

after assuming does notchange(at least not significantly)

the dependececan beintegrated:

2 1 1( 2) ( )

with 2(usually 2)

m m

m m m m m

a C K C F aN

F

a

Nm CF a a

m m

1a 2a


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Example:

1 2

14

1-2 /2 ( 2)/2 ( 2)/21 2

5

1mm, 5mm,for example 100mm

1.12100 MPa

3.25mm/cycle7.481 10

(MPa mm)

2 1 1( 2) ( )

4.620 10 cycles

m

m m m m m

a aa b bF

m

C

Nm CF a a

1a 2a

b


51Pisa, June 15 – July 14, 2015


Example:

1 2

i

ii iii iv

51-2 i ii iii iv

1mm, 5mmlimited width, 10 mmthecalculation can be donestepwisedividing the crack range in small steps:

1.38( 1 2mm)1.65, 2.1, 2.7

1.700 10 cycles

a ab

F aF F F

N N N N N

1a 2a

b

52


Example:

1 2

51-2

51-2

1mm, 5mmlimited width, 10 mmParis' law (numerical) integrationMATLAB

1.843 10 cycles

(previous result 1.700 10 cycles)

a ab

N

N

100cycles(small)stepsintegration


53


AFGROW software for crack propagation calculation


54


ASTM E647


55

ASTM standard E647

DefinitionsUsually tests are performedat positive load ratiose.g.: 0.1R


56

ASTM standard E647

Specimen geometry

For fracture toughness B = W/2, while for fatigue this requirement is less demanding.The KI values experienced by the specimen under fatigue propagation are much lower.


57

ASTM standard E647

Specimen geometry

LEFM validity condition, plane stress which is more demanding


58

ASTM standard E647

Specimen notch preparation


59

ASTM standard E647

Delta SIF calculation

Same relation to find the Stress Intensity Factor for the CT specimen

Currentvalue?

a


60

ASTM standard E647

Compliance method for

the crack size

Different positions of the crack gage clip

Here is theclipdisplacement


61Pisa, June 15 – July 14, 2015

Crack size determination

Alternative ways for accurate crack size measurement during the test

- Potential drop

- Fractomat(Crack Gage)

62Pisa, June 15 – July 14, 2015


Potential drop

Calibration procedure

2

Examplecoefficients:

26.85 9.317 0.1367X XaY Y

63


Fractomat crack gage

Crack range

As the crack propagates the foil resistance increases.Having a predefined geometry the calibration equation is already given by the manufacturer


64

0

5

10

15

20

25

ch1ch2

a' [m

m]


Possible not parallel crack propagation, especially at the beginning

Application ofFractomatat the two sides

Number of cyclesChannels 1 and 2


65

ASTM standard E647

Near threshold/

high propagation rate

Near threshold

High rate

510 mm/cycle

th

7

:d 10 mm/cycled

KaN

thIc

The final part of the curve is usually not of interestonce is knownK


66Pisa, June 15 – July 14, 2015

ASTM standard E647

Near threshold – Decreasing procedure

67

ASTM standard E647

Propagation rate calculation

A large number of cycles is recommendedbetween steps the and 1(instead of just two consecutive cycles)to have a significant, still small,crack size increment

i i


68

Data example

Q+T steel, similar to AISI 4340

Several tests with Fractomat and Potential Drop

th 9.4 MPa mK

3

3.6mm/cycle0.4 10

(MPa m)m

m

C


69

Data example

Homework:

Compare the experimental data with the alternative Paris model:

th( )m mda C K KdN

th

3

9.4 MPa m3.6

mm/cycle0.4 10(MPa m)m

Km

C

0.01

0.1

1

10

100

1000

5 50

POTENTIAL DROP

CRACK GAGE


fracture mechanics experimental tests (astm standards...

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